ese 531: digital signal processing - seas.upenn.eduese531/spring2018/handouts/lec19.pdf · "...

ESE 531: Digital Signal Processing

Lec 19: March 29, 2018 Discrete Fourier Transform, Pt 2

Penn ESE 531 Spring 2018 – Khanna Adapted from M. Lustig, EECS Berkeley

Today

!  Review: "  Discrete Fourier Transform (DFT) "  Circular Convolution

!  Fast Convolution Methods

2 Penn ESE 531 Spring 2018 - Khanna

Discrete Fourier Transform

!  The DFT

!  It is understood that,

3 Penn ESE 531 Spring 2018 – Khanna Adapted from M. Lustig, EECS Berkeley

4

DFT vs DTFT

!  Back to example


Properties of the DFS/DFT


Properties (Continued)


Duality


Proof of Duality


Discrete Fourier Series

!  Properties of WN: "  WN

0 = WNN = WN

2N = ... = 1 "  WN

k+r = WNkWN

r and, WNk+N = WN

k

!  Example: WNkn (N=6)


Proof of Duality


Circular Convolution

!  Circular Convolution:

For two signals of length N


Compute Circular Convolution Sum


Result

16

y[0]=2 y[1]=2 y[2]=3 y[3]=4


Circular Convolution

!  For x1[n] and x2[n] with length N

"  Very useful!! (for linear convolutions with DFT)


Multiplication

!  For x1[n] and x2[n] with length N


Linear Convolution

!  Next.... "  Using DFT, circular convolution is easy

"  Matrix multiplication… more later

"  But, linear convolution is useful, not circular "  So, show how to perform linear convolution with circular

convolution "  Use DFT to do linear convolution (via circular

convolution)


Linear Convolution

!  We start with two non-periodic sequences:

"  E.g. x[n] is a signal and h[n] a filter’s impulse response


Linear Convolution

!  We start with two non-periodic sequences:

"  E.g. x[n] is a signal and h[n] a filter’s impulse response

!  We want to compute the linear convolution:

"  y[n] is nonzero for 0 ≤ n ≤ L+P-2 (ie. length M=L+P-1)


Requires L*P multiplications

Linear Convolution via Circular Convolution

!  Zero-pad x[n] by P-1 zeros

!  Zero-pad h[n] by L-1 zeros

!  Now, both sequences are length M=L+P-1


Linear Convolution via Circular Convolution

!  Now, both sequences are length M=L+P-1 !  We can now compute the linear convolution using a

circular one with length M=L+P-1


Example


Linear Convolution with DFT

!  In practice we can implement a circulant convolution using the DFT property:


Linear Convolution with DFT

!  In practice we can implement a circulant convolution using the DFT property:

!  Advantage: DFT can be computed with Nlog2N complexity (FFT algorithm later!)

!  Drawback: Must wait for all the samples -- huge delay -- incompatible with real-time filtering


Block Convolution

!  Problem: "  An input signal x[n], has very long length (could be

considered infinite) "  An impulse response h[n] has length P "  We want to take advantage of DFT/FFT and compute

convolutions in blocks that are shorter than the signal

!  Approach: "  Break the signal into small blocks "  Compute convolutions "  Combine the results


Block Convolution


Overlap-Add Method

!  Decompose into non-overlapping segments

!  The input signal is the sum of segments


Example


L=11

Overlap-Add Method

!  The output is:

"  Each output segment xr[n]*h[n] is length N=L+P-1 "  h[n] has length P "  xr[n] has length L


Overlap-Add Method

!  We can compute xr[n]*h[n] using circular convolution with the DFT

!  Using the DFT: "  Zero-pad xr[n] to length N "  Zero-pad h[n] to length N and compute DFTN{hzp[n]}

"  Only need to do once


Overlap-Add Method

!  We can compute xr[n]*h[n] using circular convolution with the DFT

!  Using the DFT: "  Zero-pad xr[n] to length N "  Zero-pad h[n] to length N and compute DFTN{hzp[n]}

"  Only need to do once

"  Compute:

!  Results are of length N=L+P-1 "  Neighboring results overlap by P-1 "  Add overlaps to get final sequence


Example of Overlap-Add

36

y

y

y


L=11

L+P-1=16


37

y

y

y


L=11

L+P-1=16


38

y

y

y


L=11

L+P-1=16

Overlap-Save Method

!  Basic idea: !  Split input into overlapping segments with length

L+P-1 "  P-1 sample overlap

!  Perform circular convolution in each segment, and

keep the L sample portion which is a valid linear convolution


Example of Overlap-Save


L+P-1=16

Circular to Linear Convolution

!  An L-point sequence circularly convolved with a P-point sequence "  with L - P zeros padded, P < L

!  gives an L-point result with "  the first P - 1 values incorrect and "  the next L - P + 1 the correct linear convolution result




L+P-1=16 P-1=5 Overlap samples



L+P-1=16

Discrete Cosine Transform

!  Similar to the discrete Fourier transform (DFT), but using only real numbers

!  Widely used in lossy compression applications (eg. Mp3, JPEG)

!  Why use it?


DFT Problems

!  For processing 1-D or 2-D signals (especially coding), a common method is to divide the signal into “frames” and then apply an invertible transform to each frame that compresses the information into few coefficients.

!  The DFT has some problems when used for this purpose: "  N real x[n] ↔ N complex X[k] : 2 real, N/2 − 1 conjugate pairs "  DFT is of the periodic signal formed by replicating x[n]


DFT Problems

!  For processing 1-D or 2-D signals (especially coding), a common method is to divide the signal into “frames” and then apply an invertible transform to each frame that compresses the information into few coefficients.

!  The DFT has some problems when used for this purpose: "  N real x[n] ↔ N complex X[k] : 2 real, N/2 − 1 conjugate pairs "  DFT is of the periodic signal formed by replicating x[n] ⇒ Spurious frequency components from boundary discontinuity


The Discrete Cosine Transform (DCT) overcomes these problems.


!  To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in reverse order and insert a zero between each pair of samples:


FIR GLP: Type II




!  Take the DFT of length 4N real, symmetric, odd-sample-only sequence




!  Take the DFT of length 4N real, symmetric, odd-sample-only sequence

!  Result is real, symmetric and anti-periodic: only need first N values



!  Result is real, symmetric and anti-periodic: only need first N values


Basis Functions


DFT of Sine Wave


DCT of Sine Wave


Big Ideas

!  Discrete Fourier Transform (DFT) "  For finite signals assumed to be zero outside of defined

length "  N-point DFT is sampled DTFT at N points "  DFT properties inherited from DFS, but circular

operations!

!  Fast Convolution Methods "  Use circular convolution (i.e DFT) to perform fast linear

convolution "  Overlap-Add, Overlap-Save

!  DCT useful for frame rate compression of large signals


Admin

!  HW 8 due tomorrow !  Project

"  Work in groups of up to 2 "  Can work alone if you want "  Use Piazza to find partners


ese 531: digital signal processing - seas.upenn.eduese531/spring2018/handouts/lec19.pdf · "...

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